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Math and science::Algebra::Aluffi

Polynomial rings

When polynomials are viewed as a type of notational container, they can be seen to have a ring structure.

Polynomial

A polynomial is a combination of "powers" of an indeterminate \( x \) with coefficients in a ring \( R \). It is written in the form:

\[ a_n x^n + ... + a_1 x + a_0 \]

or

\[ \sum_{i>0} a_i n^{i} \]

We require that there is an \( n \) above which all coefficients are zero.

Two polynomials are considered equal if [what?].

The meaning of the operation \( \cdot \) in \( a_i \cdot x^i \) is not specified in the definition. Furthermore, the meaning of the \( + \) operation between elements is also not specified. The definition of equality cements the notational behaviour.

Notation

The set of polynomials with indeterminate \( x \) over ring \( R \) is denoted as \( R[x] \).

Polynomials as rings

A polynomial \( R[x] \) forms a ring (operations on the reverse side). As polynomials form rings, the construction can be nested such that polynomials in indeterminate \( y \) over \( R[x] \) for a ring. This is denoted as \( R[x][y] \) and \( R[x,y] \).

Polynomial rings can be generalized and be considered instances of [what?] rings.